On the Application of the Incomplete QR Algorithm
to the Analysis of Microstrip Antennas
Pawel GLUCHOWSKI, Andrzej A. KUCHARSKI
Faculty of Electronics,Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 53-370 Wrocław [email protected], [email protected]
Manuscript received April 1, 2015
Abstract.In this paper, we provide some insight into the us-age of fast, iterative, method-of-moments (MoM) solution of integral equations (IE) describing antennas and other metal-lic structures immersed in a planar multilayered environ-ment. Based on the form of multilayered media Green’s functions, we extract free-space terms, associated with direct rays within the analyzed structure, reducing the number of significant interactions required to describe the rest of MoM matrix. Next, we show that it is possible to construct a hybrid algorithm, where the fast multipole method (FMM) is used to the free-space matrix part, while the reduced rank incomplete QR (iQR) decomposition is applied to the remaining portion of the MoM matrix.
This HM-iQR (hybrid multipole - incomplete QR) method is applied to a relatively large (in terms o f the number of un-knowns) problem of plane wave scattering by a finite array of rectangular microstrip patches printed on a grounded di-electric slab. Computation results from the new algorithm are compared to literature data and to the results of the pure low rank IE-QR method.
Keywords
Method-of-moments, fast multipole method, low rank IE-QR algorithm, integral equations
1. Introduction
Analysis of metal structures placed in a planar multi-layered environment has been of interest to the community of researchers dealing with computational electromagnet-ics for decades. Although the integral equations/method-of-moments approach to that problem is well established [1], the issue of long computation times for electrically large objects is still relevant. The problem, as in the free-space situation, is associated with the large number of unknowns N in the final linear equation set.
In free-space problems, limitations due to the compu-tational complexity of order O(N3), when applying direct
methods like LU decomposition, are usually overcome by
us-ing Krylov-subspace iterative methods with techniques like fast multipole method (FMM) [2] or low rank IE-QR algo-rithm [3] applied to speed-up computation of matrix-vector product. Both techniques allow for reducing the cost of a single iteration from O(N2)to O(N3/2)in the case of the
single-level method, or to O(N log N) in the case of its mul-tilevel counterpart, in the case of FMM called mulmul-tilevel fast multipole algorithm (MFLMA) (cf. [2, 3]).
In multilayered media, the application of the FMM is limited, as the algorithm is based on the clever representa-tion of the free-space Green’s funcrepresenta-tion [2]. One solurepresenta-tion is to represent multilayered Green’s function with set free-space-like functions, which is called discrete complex image method (DCIM) [2, 4]. However, for structures with large lateral dimensions, the physical phenomena are often dom-inated by surface waves, for which discrete images are not well-suited [5]. On the other hand, the low rank IE-QR algo-rithm (in this paper called the incomplete QR - iQR) is based on purely algebraic MoM matrix operations, and as such -does not depend on the form of Green’s functions. However, as shown below, for some configurations it can be made faster by applying the hybridization with FMM.
The rest of this paper is organized as follows: first we formulate the representative problem concerning a finite ar-ray of rectangular microstrip patches printed on a grounded dielectric slab. Then, we outline the conventional method of analysis, as well as FMM and iQR algorithms, and the idea of the hybrid method is proposed, which enables analysis of the properties of the iQR when applied to planar stratified media problems. Finally, we show computational examples validating the proposed idea and making it possible to draw some general conclusions.
It is to be stated that by combining the two methods with computational complexity of (in the case of the single level implementation) order O(N3/2), we do not expect to obtain
algorithm with lower complexity. Instead, we expect to re-duce the constant factor standing in front of N3/2, and, what
is more important, to indicate what physical mechanisms are relevant, when applying iQR to multilayered problems.
Throughout the paper the ejωt time convention is
used.
2. Formulation
Let’s consider a finite array of microstrip patches de-picted in Fig. 1. The patches are made of the perfect electric conductor (PEC). The structure is illuminated by a plane wave, coming from the specified direction from the upper half-space, which is filled with air. The quantity to be deter-mined is the radar cross section (RCS).
The total electric field at any point of space is the su-perposition of the incident and the scattered fields. On the surface of the PEC patches, it satisfies the boundary condi-tion:
(Es(J)+ Ei)t= 0 (1)
where Eiindicates the incident field, while sub-script t stands
for tangential field component. The scattered field Esis
gen-erated by the surface current on the patches:
Es(J)= −jωA − ∇Φ, (2)
with the magnetic vector potential A and scalar potential Φ defined as: A(r)= µ0 Z S J(r0)KA(r, r0)dS0, (3) Φ(r)= −1 0 Z S ∇ · J(r0) jω K Φ (r, r0)dS0. (4) In (3) and (4) S represents total surface of the patches. Vectors r i r0denote observation and source points,
respec-tively, µ0 and 0 are permeability and permittivity of free
space, J denotes the surface current, and KA (dyadic) and
KΦ(scalar) are the Green’s functions [1], which can be
ex-pressed as Sommerfeld integrals of their spectral domain counterparts ˜KA
and ˜KΦ. For the problem at hand, the
currents are assumed planar, and also only horizontal field components are taken into account in (1). Thus, only ˜GA
vv
and ˜KΦ from the whole set described in [1] are necessary,
which are expressed by means of so-called transmission line Green’s functions (TLGFs) Vh i and Vie, as [1]: ˜GA vv= 1 jωµ0Vih, (5) ˜KΦ= −jω0 kρ2 (V h i − Vie). (6) TLGFs Vp
i for the case depicted in Fig. 1, expressed in
the notation of [1], have the general form: Vip(z|z0)= Z p n 2 e− jkz n|z−z0|+ 1 Dnp 4 X s=1 Rnsp e− jkz nγn s . (7) The first term in (7) represents the direct ray between the source and the field points, while the second term repre-sents the rays that undergo partial reflections at the upper and lower slab boundaries before reaching the observation point [1]. The details description of all coefficients used in (7) can be found in [1]. Here, we only note that all the coefficients
Fig. 1.Geometry of a finite array of rectangular microstrip patches on an infinite extent grounded dielectric slab (cf. [16]).
depend on the configuration and properties of the multilay-ered environment, frequency of interest and z coordinates of source and observation points. The index n refers to the sec-tion of the equivalent transmission line that represents layered medium in the spectral domain (cf. [1]). As the microstrip patches lie on the interface separating different media, we have a freedom of choice of the section to which the interface belongs – total formulas remain the same irrespective of that choice. So, let’s assume that the interface (which means both zand z0) belongs to the upper half-space filled with air, with which we associate index n = 0. Then, substituting (7) into (5), we get: ˜GA vv= 1 2jkz0 f e− jkz0|z−z0|+ FA(kρ, z, z0)g . (8) where kz0 = q
k02− kρ2, k0is the free-space wavenumber, and FA(kρ, z, z0)comes from the second term in (7), containing the summation.
Substituting (7) into (6), we can transform ˜KΦinto the
similar form: ˜KΦ= 1 2jkz0 f e− jkz0|z−z0|+ FΦ(k ρ, z, z0)g , (9) where again FΦ(k
ρ, z, z0)corresponds to phenomena associ-ated with layered structure of the medium under investiga-tion.
In the spacial domain, the Green’s functions become [1]:
Kx xA ( ρ)= KyyA( ρ)= S0(˜GvvA (kρ)) , (10)
KΦ( ρ) = S0(˜KΦ(kρ)) , (11)
S0( ˆf(kρ)) = 1 2π
Z ∞
0 ˆf(kρ) Jn(kρρ)kρdkρ. (12)
Above, Jn is Bessel function of order n, while ρ
de-notes distance between the source and observation points. Note that in the above formulas, we dropped out the z depen-dence, as all source and observation points are confined to the single value of z.
Putting (8), (9) into (10) and (11), respectively, and using Sommerfeld identity [6]:
1 2π Z ∞ 0 kρJ0(kρρ) e− jkz0|z−z0| 2jkz0 dkρ= 1 4π e− jk0R R , (13) where R = |r − r0|denotes distance between source (r0) and
observation (r) points, we finally get: Kx xA( ρ)= KyyA( ρ) = 1 4π e− jk0ρ ρ +S0 ( 1 2jkz0F A(k ρ) ) , (14) KΦ( ρ)= 1 4π e− jk0ρ ρ + S0 ( 1 2jkz0F Φ(k ρ) ) (15) where we bear in mind that for the analyzed case z = z0, so
Rpresent in (13) is equal to the lateral distance ρ. Details regarding integration of Sommerfeld integrals may be found in [7].
Please note that when the multilayered medium is re-placed by free-space, the corresponding transmission lines in the spectral domain become infinite ones with constant parameters. Then, FA(k
ρ, z, z0)and FΦ(k
ρ, z, z0)terms dis-appear, and the above formulas reduce to their well known free-space counterparts.
After suitable triangulation of S, the application of RWG basis functions [8], and after the usual testing pro-cedure, equation (1) is transformed into the matrix formula:
ZI= V (16)
where Z is a dense N × N impedance matrix, while I and Vare column vectors of length N describing, respectively, the approximation of the current distribution and the (tested) incident field on S.
Taking into account the form of (14) and (15), we can also conclude that in the general, multilayer case, the matrix elements may be expressed as the sum of two terms, one com-ing from the direct free-space rays, and the second associated with the presence of the multilayered structure. So, also the MoM matrix can be written as the sum of two matrices:
Z= Z0+ Zm, (17)
where Z0 is the "free-space" MoM matrix, while the
ele-ments of Zm (m for "multilayered") are computed with the
use of numerical integration of Sommerfeld integrals associ-ated with FAand FΦ. Apart from the way, how MoM matrix
entries are computed, putting Z into the form of (17) has also consequences for the solution of the linear system, which will become apparent in the subsequent discussion.
Finally, when (16) is solved, the obtained approxima-tion of the current allows to compute resulting scattered fields at arbitrary observation point. In the subsequent exam-ples we use far-field approximate formulas for multilayared medium [9].
3. HM-iQR Algorithm
3.1 FMM and Incomplete QR
As outlined in the Introduction, when an iterative method is applied to solve (16), either FMM or incomplete QR may be used to speed-up the associated matrix-vector multiply, depending on the type of the problem.
The fast multipole method is a numerical algorithm pro-posed by Greengard and Rokhlin for solving Helmholtz and Laplace problems [2, 10]. Nowadays, FMM and its multi-level extension MLFMA are applied to seek rapid solution to integral equations of scattering. The detailed description of FMM can be found for example in [10, 11], so we will omit here the detailed description.
In contrast to the above, some details about the incom-plete QR method are required to follow the idea shown in the paper, so we decided to include here a short outline. In the method, the first step is to divide the N basis functions into G localized groups, each supporting M = N/G basis functions. The good choice is G ∼ N0.5. The significant information
between two groups is limited if the distance between them is greater than their physical size [3]. Then the number of independent vectors needed for expression of interactions be-tween these two groups is much smaller than number of basis functions. The incomplete QR algorithm consists of four steps: column index selection, updating the Q matrix, row index selection, and computing the R matrix.
The algorithm begins with the arbitrary selection of a re-ceiver from the rere-ceiver group. With respect to this rere-ceiver, the most dominant transmitter is chosen from the transmit-ter group. The most dominant transmittransmit-ter will then in turn determine the most dominant receiver, which is not the du-plicate of the chosen one. With these two chosen receivers, the second most dominant transmitter is selected. This pro-cess continues until the remaining receivers and transmitters contribute no significant information [12]. In this process Modified Gram-Schmidt (MGS) algorithm is applied to de-termine the index of the column vector in Z, that is the most linearly independent on the previously selected columns. In this case local impedance matrix Zm×n
i j is factorized into
Qm×ri j and Rr ×ni j , where i and j are receiving and transmitting group, respectively, and m and n are the number of basis functions, and r is the approximate rank of Zm×n
i j [3]. This
Zm×ni j = [z1 z2 ... zn] = v1 v2 .. . vm = Qm×r i j · Rr ×ni j . (18)
Here zpis the column vector due to the pth basis
func-tion in the transmitting group, and vq is the row vector due
to the qth basis function in the receiving group [13]. This low rank IE-QR factorization reduces the complexity matrix-vector product from O(mn) to O(r(n + m)), where r is much less than m and n for well separated groups. This leads to speed up solution time for general electromagnetic radiation and scattering problems. Detailed description of incomplete QR algorithm can be found in [3].
Note that in both FMM and iQR algorithms, the speed-up is associated with well-separated grospeed-ups. Self interac-tions, and interactions with neighboring groups are computed as in the standard MoM solutions.
3.2 The Hybrid Method
Taking into account (17), we can see that each matrix-vector multiplication in the iterative algorithm may be com-puted via separate multiplications of the two matrices in-volved, namely Z0and Zm. To speed-up the first
multiplica-tion, we can use FMM, which in free space can be faster than incomplete QR [14]. For the second multiplication, the iQR is applied. Here however, this algorithm should run faster than in the case "pure" iQR method, i.e. the method with-out extraction of Z0, as the number of significant relations
between receivers and transmitters is expected to be smaller. Also, when distances between groups increase, the number of significant relations becomes smaller [15]. We check those expectations in the next section.
4. Numerical Results
In order to check the correctness and efficiency of the algorithms, they were implemented in Fortran, and run on the Intel®Xeon®X5472 3.00 GHz workstation with 64 GB of RAM. No parallelization was used, so the computations were performed on a single core.
As computational examples, we analyzed antenna ar-rays described in [16], and shown in Fig. 1. Different num-bers of patches (3 × 3, 7 × 7, and 11 × 11) were taken into account. The quantity of interest was monostatic radar cross section (RCS) for a θ polarized plane wave incident from broadside. The results obtained by iQR and HM-iQR meth-ods are compared with literature data [16, 17].
The results (RCS vs. frequency) are shown in Figs. 2, 3 and 4. It can be seen that both iQR and HM-iQR methods give results consistent with literature data, so the methods are correct. Let’s take a look at the number of significant interactions in off-diagonal MoM matrix blocks in both iQR and HM-iQR (Zmportion). In our case, the groups defined
2.0 2.5 3.0 3.5 4.0 Frequency [GHz] 35 30 25 20 15 10 5 0 5 RCS [dBsm] [16] iQR HM-iQR
Fig. 2.RCS(σθ θ) versus frequency for 3 × 3 arrays. L = 3.66 cm, W = 2.60 cm, a = b = 5.5517 cm, r= 2.17, d= 0.158 cm, θ = 0◦, φ = 45◦. 2.0 2.5 3.0 3.5 4.0 Frequency [GHz] 20 15 10 5 0 5 10 15 20 RCS [dBsm] [16] iQR HM-iQR
Fig. 3.RCS(σθ θ) versus frequency for 7 × 7 arrays. L = 3.66 cm, W = 2.60 cm, a = b = 5.5517 cm, r= 2.17, d= 0.158 cm, θ = 0◦, φ = 45◦. 2.0 2.5 3.0 3.5 4.0 Frequency [GHz] 15 10 5 0 5 10 15 20 25 RCS [dBsm] [16] iQR HM-iQR
Fig. 4.RCS(σθ θ)versus frequency for 11 × 11 arrays. L = 3.66 cm, W = 2.60 cm, a = b = 5.5517 cm, r = 2.17, d= 0.158 cm, θ = 0◦, φ = 45◦.
within the algorithms are associated with patches with the same number of basis/testing functions, so the MoM ma-trix blocks corresponding to group-group interactions are all square, and of the same size. However, the number r in (18)
Arrays Size of singleblock Size of r in iQR partiQR HM-iQR
3 × 3 341 95 65
7 × 7 225 69 47
11 × 11 133 41 26
Tab. 1.Size of r in incomplete QR.
Arrays Unknowns Time [s]iQR HM-iQR
3 × 3 3069 504 399
7 × 7 11025 22722 19614
11 × 11 16093 42861 31038
Tab. 2.Times for incomplete QR and HM-iQR - calculation for 21 frequencies.
Arrays Unknowns Time [s]iQR HM-iQR
3 × 3 3069 24 19
7 × 7 11025 1082 934
11 × 11 16093 2041 1478
Tab. 3.Times for incomplete QR and HM-iQR - calculation for single frequency.
is different, depending on the mutual position of the two in-teracting patches. Therefore, we decided to give the mean value of r, which may be used as the overall measure of the iQR efficiency (the lower r, the better). The values of r are given in Tab. 1. We can see that r, and, what follows, the number of multiplications performed in the iterative algo-rithm, for HM-iQR is about 1/3 lower than for "pure" iQR. QR decomposition is done for well separated groups, e.g. for groups 1 and 49 in 7 × 7 case with 225 basis functions in one group. In this example for HM-iQR in matrix Zmminimum
and maximum values are 1.33 · 10−9and 3.85 · 10−7,
respec-tively. For "pure" iQR minimum and maximum values are 6.47 · 10−12and 1.31 · 10−8, respectively. We noticed also,
that in this case r is 14 and 36, for HM-iQR and "pure" iQR, respectively. Those values of r are below average results presented in Tab. 1, and it is expected, because groups 1 and 49 are extremely far away from each other.
Comparison of the times (Tab. 2 and in Tab. 3) shows, that HM-iQR method may be, depending on an example, almost 30% faster. This is visible and valuable especially in calculation of RCS not only for single frequency, but for a range of frequencies (in this paper for 21 frequency points). All calculations in HM-iQR and iQR were done with toler-ance 10−10. For case 7 × 7 the proper result was achieved
slower than in others examples. If the tolerance was changed to 9.8−10 (in both methods), the HM-iQR was about 30%
faster and gave correct result. It means, that algorithm is sen-sitive to tolerance defined in conjugate gradient method.
During numerical tests, we have also found that when increasing the distance between patches beyond a certain limit, the algorithm loses its beneficial properties. This may be attributed to the fact that for far-lying patches the coupling mechanism is dominated by surface waves phenomena, so extracting the FMM part does not reduce r in the iQR, while
adding unnecessary overhead into the computation time. This issue should however be studied in more detail in future in-vestigations.
5. Conclusions
The paper presents an enhancement into MoM computations in layered media problems concerning electrically large ob-jects. The Hybrid Multipole - incomplete QR algorithm has been applied to determine RCS of a finite array of rectangu-lar microstrip patches printed on a grounded dielectric slab. The described HM-iQR algorithm allowed to get accurate and rigorous solution to the given problem and required less computation time than the pure iQR method. Although the speed-up is not spectacular, the method allows to get some additional insight into correlation between physical phenom-ena and numerical methods.
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About the Authors. . .
Paweł GŁUCHOWSKIwas born in 1983. He received his M.Sc. from Wroclaw University of Technology in 2008. His research interests include efficient algorithms in radiation and scattering problems.
Andrzej A. KUCHARSKIwas born in 1964. He received his M.Sc., Ph.D. and D.Sc. from Wroclaw University of Tech-nology in 1988, 1994, and 2001, respectively. He is the Head of the Antenna Theory and Computational Electromagnet-ics Group in the Telecommunications and TeleinformatElectromagnet-ics Department, Faculty of Electronics, Wroclaw University of Technology, Poland. His research interests include compu-tational electromagnetics in radiation and scattering prob-lems.
